Optimal. Leaf size=268 \[ \frac{2 \left (a^2-b^2\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d}+\frac{4 a b}{5 d \cot ^{\frac{5}{2}}(c+d x)}-\frac{4 a b}{d \sqrt{\cot (c+d x)}}+\frac{2 b^2}{7 d \cot ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.285769, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {3673, 3542, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{2 \left (a^2-b^2\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d}+\frac{4 a b}{5 d \cot ^{\frac{5}{2}}(c+d x)}-\frac{4 a b}{d \sqrt{\cot (c+d x)}}+\frac{2 b^2}{7 d \cot ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3673
Rule 3542
Rule 3529
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(a+b \tan (c+d x))^2}{\cot ^{\frac{5}{2}}(c+d x)} \, dx &=\int \frac{(b+a \cot (c+d x))^2}{\cot ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 b^2}{7 d \cot ^{\frac{7}{2}}(c+d x)}+\int \frac{2 a b+\left (a^2-b^2\right ) \cot (c+d x)}{\cot ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 b^2}{7 d \cot ^{\frac{7}{2}}(c+d x)}+\frac{4 a b}{5 d \cot ^{\frac{5}{2}}(c+d x)}+\int \frac{a^2-b^2-2 a b \cot (c+d x)}{\cot ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 b^2}{7 d \cot ^{\frac{7}{2}}(c+d x)}+\frac{4 a b}{5 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\int \frac{-2 a b-\left (a^2-b^2\right ) \cot (c+d x)}{\cot ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 b^2}{7 d \cot ^{\frac{7}{2}}(c+d x)}+\frac{4 a b}{5 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{4 a b}{d \sqrt{\cot (c+d x)}}+\int \frac{-a^2+b^2+2 a b \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx\\ &=\frac{2 b^2}{7 d \cot ^{\frac{7}{2}}(c+d x)}+\frac{4 a b}{5 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{4 a b}{d \sqrt{\cot (c+d x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{a^2-b^2-2 a b x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{d}\\ &=\frac{2 b^2}{7 d \cot ^{\frac{7}{2}}(c+d x)}+\frac{4 a b}{5 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{4 a b}{d \sqrt{\cot (c+d x)}}+\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{d}+\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{d}\\ &=\frac{2 b^2}{7 d \cot ^{\frac{7}{2}}(c+d x)}+\frac{4 a b}{5 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{4 a b}{d \sqrt{\cot (c+d x)}}+\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 d}+\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 d}-\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 \sqrt{2} d}-\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 \sqrt{2} d}\\ &=\frac{2 b^2}{7 d \cot ^{\frac{7}{2}}(c+d x)}+\frac{4 a b}{5 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{4 a b}{d \sqrt{\cot (c+d x)}}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} d}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} d}+\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}-\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}\\ &=-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{2 b^2}{7 d \cot ^{\frac{7}{2}}(c+d x)}+\frac{4 a b}{5 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{4 a b}{d \sqrt{\cot (c+d x)}}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} d}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} d}\\ \end{align*}
Mathematica [C] time = 0.392241, size = 80, normalized size = 0.3 \[ \frac{2 \left (a \left (5 a+14 b \cot (c+d x) \, _2F_1\left (-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(c+d x)\right )\right )-5 \left (a^2-b^2\right ) \, _2F_1\left (-\frac{7}{4},1;-\frac{3}{4};-\cot ^2(c+d x)\right )\right )}{35 d \cot ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.362, size = 1903, normalized size = 7.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.69309, size = 302, normalized size = 1.13 \begin{align*} \frac{8 \,{\left (15 \, b^{2} + \frac{42 \, a b}{\tan \left (d x + c\right )} - \frac{210 \, a b}{\tan \left (d x + c\right )^{3}} + \frac{35 \,{\left (a^{2} - b^{2}\right )}}{\tan \left (d x + c\right )^{2}}\right )} \tan \left (d x + c\right )^{\frac{7}{2}} + 210 \, \sqrt{2}{\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + \frac{2}{\sqrt{\tan \left (d x + c\right )}}\right )}\right ) + 210 \, \sqrt{2}{\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - \frac{2}{\sqrt{\tan \left (d x + c\right )}}\right )}\right ) + 105 \, \sqrt{2}{\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\tan \left (d x + c\right )} + 1\right ) - 105 \, \sqrt{2}{\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\tan \left (d x + c\right )} + 1\right )}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}{\cot \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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